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- Arvesen, J. N. 1969. Jackknifing U-statistics.Annals of Mathematical Statistics40: 2076–2100.

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- inside the square brackets are kernels of these regular functionals.) The quantities we really want to estimate are Kendall’s-aand Somers’D, defined respectively by -XY=TXY=V;DYX=TXY=TXX=-XY=-XX(10) (These are equal to the familiar formulas (1) and (2) if each cluster contains one observation with an importance weight of one.) To estimate these, we use the jackknife method of
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- Arvesen (1969)
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- on the regular functionals (9) and use appropriate Taylor polynomials. The functionalsVandTXYare estimated by the Hoeffding (1948)U-statistics Vb=v:::: n(n-1) ;bTXY= t(XY):::: n(n-1) (11) and the respective jackknife pseudovalues corresponding to thehth cluster are given by (V)h=(n-1)-1v::::-(n-2)-1[v::::-2vh:::] (XY) h=(n-1) -1t(XY) ::::-(n-2) -1 h t(XY)::::-2t (XY) h::: i (12) somers

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- Daniels, H. E. and M. G. Kendall. 1947. The significance of rank correlation where parental correlation exists.Biometrika34: 197–208.

Total in-text references: 1- In-text reference with the coordinate start=154491
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- parameter and its covariance matrix are -b=-(b-);bC(-)=b-(-)bC(-)b-(-)0(19) Fisher’sz-transform was originally recommended for the Pearson correlation coefficient by Fisher (1921) (see also Gayen 1951), but Edwardes (1995) recommended it specifically for Somers’Don the basis of simulation studies. Daniels’ arcsine was suggested as a normalizing transform in
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- Daniels and Kendall (1947)
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- Iftransf(z)ortransf(asin)is specified, then somersdprints asymmetric confidence intervals for the untransformedDor-avalues, calculated from symmetric confidence intervals for the transformed parameters using the inverse function-(-).

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- Edwardes M. D. deB. 1995. A confidence interval for Pr(X<Y)-Pr(X>Y) estimated from simple cluster samples.Biometrics51: 571–578.

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- estimates given by (18), andb-(-)is the diagonal matrix whose diagonal entries are thed-=d-estimates specified in the table, then the transformed parameter and its covariance matrix are -b=-(b-);bC(-)=b-(-)bC(-)b-(-)0(19) Fisher’sz-transform was originally recommended for the Pearson correlation coefficient by Fisher (1921) (see also Gayen 1951), but
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- Edwardes (1995)
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- recommended it specifically for Somers’Don the basis of simulation studies. Daniels’ arcsine was suggested as a normalizing transform in Daniels and Kendall (1947). Iftransf(z)ortransf(asin)is specified, then somersdprints asymmetric confidence intervals for the untransformedDor-avalues, calculated from symmetric confidence intervals for the transformed parameters

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- are mutually complementary.) Weight (lbs.) 1,0002,0003,0004,0005,000 0 5 10 15 20 25 30 35 40 45 F D D FF F F F F F F DD F F D F DF FF F D DDD F F DDD FD F D F D DD DD D DDD DDDDDD F DDD D D D DD D D D D D D D D DD D D DD Figure 1. Applyingsomersdto theautodata. The confidence intervals for such high values of Somers’Dwould probably be more reliable if we used thez-transform, recommended by
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- Edwardes (1995)
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- The results of this are as follows: .somersdforeignmpgweight,tran(z) Somers'D Transformation:Fisher'sz Validobservations:74 -----------------------------------------------------------------------------|Jackknife foreign|Coef.

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- Fisher, R. A. 1921. On the “probable error” of a coefficient of correlation deduced from a small sample.Metron1(4): 3–32.

Total in-text references: 1- In-text reference with the coordinate start=154273
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- (-)is the covariance matrix for the untransformed estimates given by (18), andb-(-)is the diagonal matrix whose diagonal entries are thed-=d-estimates specified in the table, then the transformed parameter and its covariance matrix are -b=-(b-);bC(-)=b-(-)bC(-)b-(-)0(19) Fisher’sz-transform was originally recommended for the Pearson correlation coefficient by
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- Fisher (1921)
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- (see also Gayen 1951), but Edwardes (1995) recommended it specifically for Somers’Don the basis of simulation studies. Daniels’ arcsine was suggested as a normalizing transform in Daniels and Kendall (1947).

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- Gayen, A. K. 1951. The frequency distribution of the product-moment correlation coefficient in random samples of any size drawn from non-normal universes.Biometrika38: 219–247.

Total in-text references: 1- In-text reference with the coordinate start=154301
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- for the untransformed estimates given by (18), andb-(-)is the diagonal matrix whose diagonal entries are thed-=d-estimates specified in the table, then the transformed parameter and its covariance matrix are -b=-(b-);bC(-)=b-(-)bC(-)b-(-)0(19) Fisher’sz-transform was originally recommended for the Pearson correlation coefficient by Fisher (1921) (see also
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- Gayen 1951)
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- but Edwardes (1995) recommended it specifically for Somers’Don the basis of simulation studies. Daniels’ arcsine was suggested as a normalizing transform in Daniels and Kendall (1947).

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- Hoeffding, W. 1948. A class of statistics with asymptotically normal distribution.Annals of Mathematical Statistics19: 293–325.

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- notation to define (for instance) vh:j:= Xmh i=1 Xmj k=1 vhijk;t(XY)h:j:= Xmh i=1 Xmj k=1 t(XY)hijk;vh:::= Xn j=1 vh:j:;t(XY)h:::= Xn j=1 t(XY)h:j:(8) and any other sums over any other indices. Given that the clusters are sampled independently from a common population of clusters, we can define V=E[vh:j:];TXY=E h t(XY)h:j: i (9) for allh6=j. (In the terminology of
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- Hoeffding (1948)
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- these quantities are regular functionals of the cluster population distribution, and the expressions inside the square brackets are kernels of these regular functionals.) The quantities we really want to estimate are Kendall’s-aand Somers’D, defined respectively by -XY=TXY=V;DYX=TXY=TXX=-XY=-XX(10) (These are equal to the familiar formulas (1) and (2) if each cluster contains one observation with

- In-text reference with the coordinate start=151234
- Prefix
- Somers’D, defined respectively by -XY=TXY=V;DYX=TXY=TXX=-XY=-XX(10) (These are equal to the familiar formulas (1) and (2) if each cluster contains one observation with an importance weight of one.) To estimate these, we use the jackknife method of Arvesen (1969) on the regular functionals (9) and use appropriate Taylor polynomials. The functionalsVandTXYare estimated by the
- Exact
- Hoeffding (1948)
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- U-statistics Vb=v:::: n(n-1) ;bTXY= t(XY):::: n(n-1) (11) and the respective jackknife pseudovalues corresponding to thehth cluster are given by (V)h=(n-1)-1v::::-(n-2)-1[v::::-2vh:::] (XY) h=(n-1) -1t(XY) ::::-(n-2) -1 h t(XY)::::-2t (XY) h::: i (12) somersdcalculates correlation measures for a single variableXwith a set ofY-variates(Y(1);:::;Y(p)).

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- Kendall, M. G. 1949. Rank and product-moment correlation.Biometrika36: 177–193. ——. 1970.Rank Correlation Methods. 4th ed. London: Griffin.

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- Under this relation, Kendall’s-a-values of 0,-13,-12 and-1correspond to Pearson’s correlations of 0,-12,-1p2and-1, respectively. A similar correspondence is likely to hold in a wider range of continuous bivariate distributions
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- (Kendall 1949, Newson 1987)
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- Newson 1987). 3. Kendall’s-ahas the desirable property that a larger-acannot be secondary to a smaller-a, that is, if a positive-XYis caused entirely by a monotonic positive relationship of both variables with a third variableW, then-WXand-WYmust both be greater than-XY.

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- Newson, R. B. 1987. An analysis of cinematographic cell division data using U-statistics [Dphil dissertation]. Brighton, UK: Sussex University, 301–310.

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- Under this relation, Kendall’s-a-values of 0,-13,-12 and-1correspond to Pearson’s correlations of 0,-12,-1p2and-1, respectively. A similar correspondence is likely to hold in a wider range of continuous bivariate distributions (Kendall 1949,
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- Newson 1987)
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- . Kendall’s-ahas the desirable property that a larger-acannot be secondary to a smaller-a, that is, if a positive-XYis caused entirely by a monotonic positive relationship of both variables with a third variableW, then-WXand-WYmust both be greater than-XY.

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- Somers, R. H. 1962. A new asymmetric measure of association for ordinal variables.American Sociological Review27: 799–811. zz10Cumulative index for STB-49–STB-54 [an] Announcements

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- Remarks The population value of Kendall’s-a(Kendall 1970) is defined as -XY=E[sign(X1-X2)sign(Y1-Y2)](1) where(X1;Y1)and(X2;Y2)are bivariate random variables sampled independently from the same population, andE[-]denotes expectation. The population value of Somers’D
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- (Somers 1962)
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- is defined as DYX= -XY -XX (2) Therefore,-XYis the difference between two probabilities, namely the probability that the larger of the twoX-values is associated with the larger of the twoY-values and the probability that the largerX-value is associated with the smallerY-value.

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